Rust f32::asinh

Percentage Accurate: 37.7% → 99.4%

Time: 8.5s

Alternatives: 12

Speedup: 4.0×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary32 (asinh x))

C

float code(float x) {
	return asinhf(x);
}

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Initial Program: 37.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

float code(float x) {
	return copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
}

Julia

function code(x)
	return copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + single(1.0))))));
end

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, 0.075 \cdot {x}^{5}\right), 1, x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (fma
         (fma -0.16666666666666666 (pow x 3.0) (* 0.075 (pow x 5.0)))
         1.0
         x)
        x)
       (copysign (log1p (+ -1.0 (+ x (hypot 1.0 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf(fmaf(fmaf(-0.16666666666666666f, powf(x, 3.0f), (0.075f * powf(x, 5.0f))), 1.0f, x), x);
	} else {
		tmp = copysignf(log1pf((-1.0f + (x + hypotf(1.0f, x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(fma(fma(Float32(-0.16666666666666666), (x ^ Float32(3.0)), Float32(Float32(0.075) * (x ^ Float32(5.0)))), Float32(1.0), x), x);
	else
		tmp = copysign(log1p(Float32(Float32(-1.0) + Float32(x + hypot(Float32(1.0), x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -1

   1. Initial program 56.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 8.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 8.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 8.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 11.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 11.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.0500000007

   1. Initial program 24.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 24.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 24.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 24.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 8.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 8.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 24.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 24.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 24.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 24.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 24.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)\right)} + 0, x\right) \]
   5. Step-by-step derivation

      1. *-un-lft-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{1 \cdot \left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)\right)} + 0, x\right) \]

      2. associate-+r+ 100.0%

         \[\leadsto \mathsf{copysign}\left(1 \cdot \color{blue}{\left(\left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right) + x\right)} + 0, x\right) \]

      3. distribute-rgt-in 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(\left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right) \cdot 1 + x \cdot 1\right)} + 0, x\right) \]

      4. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(\left(\left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right) \cdot 1 + \color{blue}{1 \cdot x}\right) + 0, x\right) \]

      5. *-un-lft-identity 100.0%

         \[\leadsto \mathsf{copysign}\left(\left(\left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}\right) \cdot 1 + \color{blue}{x}\right) + 0, x\right) \]

      6. fma-def 100.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot {x}^{3} + 0.075 \cdot {x}^{5}, 1, x\right)} + 0, x\right) \]

      7. fma-def 100.0%

         \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, 0.075 \cdot {x}^{5}\right)}, 1, x\right) + 0, x\right) \]

   6. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, 0.075 \cdot {x}^{5}\right), 1, x\right)} + 0, x\right) \]

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 57.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. log1p-expm1-u 57.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1\right), x\right) \]

      4. add-sqr-sqrt 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) - 1\right), x\right) \]

      5. fabs-sqr 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) - 1\right), x\right) \]

      6. add-sqr-sqrt 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) - 1\right), x\right) \]

      7. +-commutative 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) - 1\right), x\right) \]

      8. hypot-1-def 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) - 1\right), x\right) \]

   3. Applied egg-rr 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, {x}^{3}, 0.075 \cdot {x}^{5}\right), 1, x\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right)\\ \mathbf{if}\;t_0 \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;t_0 \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x)))
   (if (<= t_0 -1.0)
     (copysign (- (log (- (hypot 1.0 x) x))) x)
     (if (<= t_0 0.05000000074505806)
       (copysign
        (+ (* -0.16666666666666666 (pow x 3.0)) (+ x (* 0.075 (pow x 5.0))))
        x)
       (copysign (log1p (+ -1.0 (+ x (hypot 1.0 x)))) x)))))

C

float code(float x) {
	float t_0 = copysignf(logf((fabsf(x) + sqrtf(((x * x) + 1.0f)))), x);
	float tmp;
	if (t_0 <= -1.0f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (t_0 <= 0.05000000074505806f) {
		tmp = copysignf(((-0.16666666666666666f * powf(x, 3.0f)) + (x + (0.075f * powf(x, 5.0f)))), x);
	} else {
		tmp = copysignf(log1pf((-1.0f + (x + hypotf(1.0f, x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = copysign(log(Float32(abs(x) + sqrt(Float32(Float32(x * x) + Float32(1.0))))), x)
	tmp = Float32(0.0)
	if (t_0 <= Float32(-1.0))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (t_0 <= Float32(0.05000000074505806))
		tmp = copysign(Float32(Float32(Float32(-0.16666666666666666) * (x ^ Float32(3.0))) + Float32(x + Float32(Float32(0.075) * (x ^ Float32(5.0))))), x);
	else
		tmp = copysign(log1p(Float32(Float32(-1.0) + Float32(x + hypot(Float32(1.0), x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < -1

   1. Initial program 56.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 8.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 8.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 8.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 11.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 11.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -1 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x) < 0.0500000007

   1. Initial program 24.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 24.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 24.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 24.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 8.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 8.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 24.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 24.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 24.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 24.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 24.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(-0.16666666666666666 \cdot {x}^{3} + \left(0.075 \cdot {x}^{5} + x\right)\right)} + 0, x\right) \]

   if 0.0500000007 < (copysign.f32 (log.f32 (+.f32 (fabs.f32 x) (sqrt.f32 (+.f32 (*.f32 x x) 1)))) x)

   1. Initial program 57.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. log1p-expm1-u 57.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1\right), x\right) \]

      4. add-sqr-sqrt 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) - 1\right), x\right) \]

      5. fabs-sqr 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) - 1\right), x\right) \]

      6. add-sqr-sqrt 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) - 1\right), x\right) \]

      7. +-commutative 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) - 1\right), x\right) \]

      8. hypot-1-def 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) - 1\right), x\right) \]

   3. Applied egg-rr 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq -1:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-0.16666666666666666 \cdot {x}^{3} + \left(x + 0.075 \cdot {x}^{5}\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.05000000074505806)
     (copysign (- x (* (pow x 3.0) 0.16666666666666666)) x)
     (copysign (log1p (+ -1.0 (+ x (hypot 1.0 x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05000000074505806f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.05000000074505806f) {
		tmp = copysignf((x - (powf(x, 3.0f) * 0.16666666666666666f)), x);
	} else {
		tmp = copysignf(log1pf((-1.0f + (x + hypotf(1.0f, x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.05000000074505806))
		tmp = copysign(Float32(x - Float32((x ^ Float32(3.0)) * Float32(0.16666666666666666))), x);
	else
		tmp = copysign(log1p(Float32(Float32(-1.0) + Float32(x + hypot(Float32(1.0), x)))), x);
	end
	return tmp
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 58.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 11.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 11.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 11.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 11.3%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 11.2%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 12.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 12.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 13.6%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 13.6%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 13.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 13.6%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -0.0500000007 < x < 0.0500000007

   1. Initial program 23.3%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 23.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 23.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 23.2%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 23.2%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 23.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt 8.2%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr 8.2%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 22.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 22.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 23.2%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 23.2%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 23.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

   4. Taylor expanded in x around 0 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto \mathsf{copysign}\left(-\left(0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(-x\right)}\right), x\right) \]

      2. unsub-neg 100.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} - x\right)}, x\right) \]

      3. *-commutative 100.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\color{blue}{{x}^{3} \cdot 0.16666666666666666} - x\right), x\right) \]

   6. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left({x}^{3} \cdot 0.16666666666666666 - x\right)}, x\right) \]

   if 0.0500000007 < x

   1. Initial program 57.7%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. log1p-expm1-u 57.7%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)\right)}, x\right) \]

      2. expm1-udef 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1}\right), x\right) \]

      3. add-exp-log 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)} - 1\right), x\right) \]

      4. add-sqr-sqrt 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) - 1\right), x\right) \]

      5. fabs-sqr 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) - 1\right), x\right) \]

      6. add-sqr-sqrt 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) - 1\right), x\right) \]

      7. +-commutative 57.7%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) - 1\right), x\right) \]

      8. hypot-1-def 99.9%

         \[\leadsto \mathsf{copysign}\left(\mathsf{log1p}\left(\left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) - 1\right), x\right) \]

   3. Applied egg-rr 99.9%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{log1p}\left(\left(x + \mathsf{hypot}\left(1, x\right)\right) - 1\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\mathsf{log1p}\left(-1 + \left(x + \mathsf{hypot}\left(1, x\right)\right)\right), x\right)\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.800000011920929:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -0.05000000074505806)
   (copysign (- (log (- (hypot 1.0 x) x))) x)
   (if (<= x 0.800000011920929)
     (copysign (- x (* (pow x 3.0) 0.16666666666666666)) x)
     (copysign (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -0.05000000074505806f) {
		tmp = copysignf(-logf((hypotf(1.0f, x) - x)), x);
	} else if (x <= 0.800000011920929f) {
		tmp = copysignf((x - (powf(x, 3.0f) * 0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf(((x * 2.0f) + (0.5f * (1.0f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-0.05000000074505806))
		tmp = copysign(Float32(-log(Float32(hypot(Float32(1.0), x) - x))), x);
	elseif (x <= Float32(0.800000011920929))
		tmp = copysign(Float32(x - Float32((x ^ Float32(3.0)) * Float32(0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-0.05000000074505806))
		tmp = sign(x) * abs(-log((hypot(single(1.0), x) - x)));
	elseif (x <= single(0.800000011920929))
		tmp = sign(x) * abs((x - ((x ^ single(3.0)) * single(0.16666666666666666))));
	else
		tmp = sign(x) * abs(log(((x * single(2.0)) + (single(0.5) * (single(1.0) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -0.0500000007

   1. Initial program 58.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 11.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 11.3%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 11.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 11.3%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 11.2%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 12.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 12.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 13.6%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 13.6%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 13.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 13.6%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 99.8%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   if -0.0500000007 < x < 0.800000012

   1. Initial program 23.8%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 23.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 23.6%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 23.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 23.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 23.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt 8.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr 8.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 23.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 23.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 23.8%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 23.8%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 24.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

   4. Taylor expanded in x around 0 99.5%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 99.5%

         \[\leadsto \mathsf{copysign}\left(-\left(0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(-x\right)}\right), x\right) \]

      2. unsub-neg 99.5%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} - x\right)}, x\right) \]

      3. *-commutative 99.5%

         \[\leadsto \mathsf{copysign}\left(-\left(\color{blue}{{x}^{3} \cdot 0.16666666666666666} - x\right), x\right) \]

   6. Simplified 99.5%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left({x}^{3} \cdot 0.16666666666666666 - x\right)}, x\right) \]

   if 0.800000012 < x

   1. Initial program 57.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 57.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 57.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 57.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 57.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 57.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 57.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 57.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around inf 99.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.05000000074505806:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\mathsf{hypot}\left(1, x\right) - x\right), x\right)\\ \mathbf{elif}\;x \leq 0.800000011920929:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \]

Alternative 5: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.800000011920929:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (- (* x -2.0) (/ 0.5 x)))) x)
   (if (<= x 0.800000011920929)
     (copysign (- x (* (pow x 3.0) 0.16666666666666666)) x)
     (copysign (log (+ (* x 2.0) (* 0.5 (/ 1.0 x)))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf(((x * -2.0f) - (0.5f / x))), x);
	} else if (x <= 0.800000011920929f) {
		tmp = copysignf((x - (powf(x, 3.0f) * 0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf(((x * 2.0f) + (0.5f * (1.0f / x)))), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(Float32(x * Float32(-2.0)) - Float32(Float32(0.5) / x)))), x);
	elseif (x <= Float32(0.800000011920929))
		tmp = copysign(Float32(x - Float32((x ^ Float32(3.0)) * Float32(0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(Float32(x * Float32(2.0)) + Float32(Float32(0.5) * Float32(Float32(1.0) / x)))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log(((x * single(-2.0)) - (single(0.5) / x))));
	elseif (x <= single(0.800000011920929))
		tmp = sign(x) * abs((x - ((x ^ single(3.0)) * single(0.16666666666666666))));
	else
		tmp = sign(x) * abs(log(((x * single(2.0)) + (single(0.5) * (single(1.0) / x)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 56.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 8.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 8.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 8.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 11.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 11.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   8. Taylor expanded in x around -inf 98.7%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]
   9. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot -2} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      2. associate-*r/ 98.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot -2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right), x\right) \]

      3. metadata-eval 98.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{\color{blue}{0.5}}{x}\right), x\right) \]

   10. Simplified 98.7%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2 - \frac{0.5}{x}\right)}, x\right) \]

   if -2 < x < 0.800000012

   1. Initial program 24.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 24.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 24.7%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 24.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 24.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 24.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt 8.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr 8.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 24.6%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 24.6%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 24.8%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 24.8%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 25.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

   4. Taylor expanded in x around 0 99.3%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 99.3%

         \[\leadsto \mathsf{copysign}\left(-\left(0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(-x\right)}\right), x\right) \]

      2. unsub-neg 99.3%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} - x\right)}, x\right) \]

      3. *-commutative 99.3%

         \[\leadsto \mathsf{copysign}\left(-\left(\color{blue}{{x}^{3} \cdot 0.16666666666666666} - x\right), x\right) \]

   6. Simplified 99.3%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left({x}^{3} \cdot 0.16666666666666666 - x\right)}, x\right) \]

   if 0.800000012 < x

   1. Initial program 57.1%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 57.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 57.1%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 57.1%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 57.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 57.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 57.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 57.1%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around inf 99.0%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 0.800000011920929:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2 + 0.5 \cdot \frac{1}{x}\right), x\right)\\ \end{array} \]

Alternative 6: 97.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0)
     (copysign (- x (* (pow x 3.0) 0.16666666666666666)) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x - (powf(x, 3.0f) * 0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x - Float32((x ^ Float32(3.0)) * Float32(0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x - ((x ^ single(3.0)) * single(0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 56.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 56.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 56.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 56.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 14.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 14.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 14.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 14.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 14.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around -inf 96.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} + 0, x\right) \]

   if -2 < x < 1

   1. Initial program 25.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 25.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 25.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 25.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 25.3%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 25.2%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt 9.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr 9.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 25.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 25.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 25.3%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 25.3%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 25.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 98.9%

         \[\leadsto \mathsf{copysign}\left(-\left(0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(-x\right)}\right), x\right) \]

      2. unsub-neg 98.9%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} - x\right)}, x\right) \]

      3. *-commutative 98.9%

         \[\leadsto \mathsf{copysign}\left(-\left(\color{blue}{{x}^{3} \cdot 0.16666666666666666} - x\right), x\right) \]

   6. Simplified 98.9%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left({x}^{3} \cdot 0.16666666666666666 - x\right)}, x\right) \]

   if 1 < x

   1. Initial program 56.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 56.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 56.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 56.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around inf 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x\right)} + 0, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]

   6. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 7: 98.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (- (* x -2.0) (/ 0.5 x)))) x)
   (if (<= x 1.0)
     (copysign (- x (* (pow x 3.0) 0.16666666666666666)) x)
     (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf(((x * -2.0f) - (0.5f / x))), x);
	} else if (x <= 1.0f) {
		tmp = copysignf((x - (powf(x, 3.0f) * 0.16666666666666666f)), x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(Float32(x * Float32(-2.0)) - Float32(Float32(0.5) / x)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(Float32(x - Float32((x ^ Float32(3.0)) * Float32(0.16666666666666666))), x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log(((x * single(-2.0)) - (single(0.5) / x))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs((x - ((x ^ single(3.0)) * single(0.16666666666666666))));
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 56.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 8.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 8.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 8.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 11.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 11.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   8. Taylor expanded in x around -inf 98.7%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x - 0.5 \cdot \frac{1}{x}\right)}, x\right) \]
   9. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\color{blue}{x \cdot -2} - 0.5 \cdot \frac{1}{x}\right), x\right) \]

      2. associate-*r/ 98.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot -2 - \color{blue}{\frac{0.5 \cdot 1}{x}}\right), x\right) \]

      3. metadata-eval 98.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{\color{blue}{0.5}}{x}\right), x\right) \]

   10. Simplified 98.7%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2 - \frac{0.5}{x}\right)}, x\right) \]

   if -2 < x < 1

   1. Initial program 25.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 25.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 25.2%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 25.3%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 25.3%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 25.2%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt 9.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr 9.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 25.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 25.1%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 25.3%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 25.3%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 25.5%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

   4. Taylor expanded in x around 0 98.9%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} + -1 \cdot x\right)}, x\right) \]
   5. Step-by-step derivation

      1. neg-mul-1 98.9%

         \[\leadsto \mathsf{copysign}\left(-\left(0.16666666666666666 \cdot {x}^{3} + \color{blue}{\left(-x\right)}\right), x\right) \]

      2. unsub-neg 98.9%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(0.16666666666666666 \cdot {x}^{3} - x\right)}, x\right) \]

      3. *-commutative 98.9%

         \[\leadsto \mathsf{copysign}\left(-\left(\color{blue}{{x}^{3} \cdot 0.16666666666666666} - x\right), x\right) \]

   6. Simplified 98.9%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left({x}^{3} \cdot 0.16666666666666666 - x\right)}, x\right) \]

   if 1 < x

   1. Initial program 56.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 56.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 56.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 56.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around inf 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x\right)} + 0, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]

   6. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2 - \frac{0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x - {x}^{3} \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 8: 97.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 1.0) (copysign x x) (copysign (- (log (/ 0.5 x))) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(-logf((0.5f / x)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(Float32(-log(Float32(Float32(0.5) / x))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(-log((single(0.5) / x)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 56.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 8.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 8.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 8.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 11.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 11.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   8. Taylor expanded in x around -inf 96.7%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x\right)}, x\right) \]
   9. Step-by-step derivation

      1. *-commutative 96.7%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   10. Simplified 96.7%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -2 < x < 1

   1. Initial program 25.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 25.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 25.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 25.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 9.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 9.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 25.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around 0 96.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + 0, x\right) \]

   if 1 < x

   1. Initial program 56.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 7.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 7.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 7.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 7.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt 8.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr 8.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 7.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 7.5%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 7.3%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 7.3%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 7.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 7.3%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- 2.7%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative 2.7%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- 2.7%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef 2.7%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative 2.7%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 13.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   8. Taylor expanded in x around inf 98.4%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\frac{0.5}{x}\right)}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(\frac{0.5}{x}\right), x\right)\\ \end{array} \]

Alternative 9: 97.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (- (log (* x -2.0))) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 56.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 8.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 8.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 8.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 11.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 11.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   8. Taylor expanded in x around -inf 96.7%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x\right)}, x\right) \]
   9. Step-by-step derivation

      1. *-commutative 96.7%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   10. Simplified 96.7%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -2 < x < 1

   1. Initial program 25.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 25.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 25.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 25.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 9.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 9.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 25.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around 0 96.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + 0, x\right) \]

   if 1 < x

   1. Initial program 56.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 56.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 56.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 56.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around inf 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x\right)} + 0, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]

   6. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 10: 97.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0)
   (copysign (log (/ -0.5 x)) x)
   (if (<= x 1.0) (copysign x x) (copysign (log (* x 2.0)) x))))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(logf((-0.5f / x)), x);
	} else if (x <= 1.0f) {
		tmp = copysignf(x, x);
	} else {
		tmp = copysignf(logf((x * 2.0f)), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(log(Float32(Float32(-0.5) / x)), x);
	elseif (x <= Float32(1.0))
		tmp = copysign(x, x);
	else
		tmp = copysign(log(Float32(x * Float32(2.0))), x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(log((single(-0.5) / x)));
	elseif (x <= single(1.0))
		tmp = sign(x) * abs(x);
	else
		tmp = sign(x) * abs(log((x * single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 3 regimes

2. if x < -2

   1. Initial program 56.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 56.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 56.9%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 56.9%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 14.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 14.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 14.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 14.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 14.3%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around -inf 96.7%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{-0.5}{x}\right)} + 0, x\right) \]

   if -2 < x < 1

   1. Initial program 25.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 25.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 25.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 25.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 9.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 9.5%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 25.6%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 25.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around 0 96.6%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + 0, x\right) \]

   if 1 < x

   1. Initial program 56.4%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 56.4%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 56.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 56.3%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 56.4%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 100.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 100.0%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around inf 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(2 \cdot x\right)} + 0, x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.5%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]

   6. Simplified 98.5%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(x \cdot 2\right)} + 0, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(\frac{-0.5}{x}\right), x\right)\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(\log \left(x \cdot 2\right), x\right)\\ \end{array} \]

Alternative 11: 75.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (if (<= x -2.0) (copysign (- (log (* x -2.0))) x) (copysign x x)))

C

float code(float x) {
	float tmp;
	if (x <= -2.0f) {
		tmp = copysignf(-logf((x * -2.0f)), x);
	} else {
		tmp = copysignf(x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = Float32(0.0)
	if (x <= Float32(-2.0))
		tmp = copysign(Float32(-log(Float32(x * Float32(-2.0)))), x);
	else
		tmp = copysign(x, x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = single(0.0);
	if (x <= single(-2.0))
		tmp = sign(x) * abs(-log((x * single(-2.0))));
	else
		tmp = sign(x) * abs(x);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if x < -2

   1. Initial program 56.9%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. flip-+ 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}{\left|x\right| - \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      2. clear-num 8.8%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\frac{1}{\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\right)}, x\right) \]

      3. log-rec 8.8%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{\left|x\right| - \sqrt{x \cdot x + 1}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right)}, x\right) \]

      4. +-commutative 8.8%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \sqrt{\color{blue}{1 + x \cdot x}}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      5. hypot-1-def 8.7%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|x\right| - \color{blue}{\mathsf{hypot}\left(1, x\right)}}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      6. add-sqr-sqrt -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      7. fabs-sqr -0.0%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      8. add-sqr-sqrt 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{\color{blue}{x} - \mathsf{hypot}\left(1, x\right)}{\left|x\right| \cdot \left|x\right| - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      9. sqr-abs 9.9%

         \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\color{blue}{x \cdot x} - \sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}\right), x\right) \]

      10. add-sqr-sqrt 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(x \cdot x + 1\right)}}\right), x\right) \]

      11. +-commutative 11.1%

          \[\leadsto \mathsf{copysign}\left(-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{x \cdot x - \color{blue}{\left(1 + x \cdot x\right)}}\right), x\right) \]

   3. Applied egg-rr 11.1%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{-\log \left(\frac{x - \mathsf{hypot}\left(1, x\right)}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]
   4. Step-by-step derivation

      1. div-inv 11.1%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(\left(x - \mathsf{hypot}\left(1, x\right)\right) \cdot \frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)}, x\right) \]

      2. log-prod -0.0%

         \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]

   5. Applied egg-rr -0.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\mathsf{fma}\left(x, x, -1\right) - x \cdot x}\right)\right)}, x\right) \]
   6. Step-by-step derivation

      1. fma-udef -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(x \cdot x + -1\right)} - x \cdot x}\right)\right), x\right) \]

      2. associate-+r- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{x \cdot x + \left(-1 - x \cdot x\right)}}\right)\right), x\right) \]

      3. +-commutative -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{\left(-1 - x \cdot x\right) + x \cdot x}}\right)\right), x\right) \]

      4. associate-+l- -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1 - \left(x \cdot x - x \cdot x\right)}}\right)\right), x\right) \]

      5. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{-1 - \color{blue}{0}}\right)\right), x\right) \]

      6. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\frac{1}{\color{blue}{-1}}\right)\right), x\right) \]

      7. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{-1}\right), x\right) \]

      8. metadata-eval -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(-1 - 0\right)}\right), x\right) \]

      9. +-inverses -0.0%

         \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(-1 - \color{blue}{\left(x \cdot x - x \cdot x\right)}\right)\right), x\right) \]

      10. associate-+l- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(-1 - x \cdot x\right) + x \cdot x\right)}\right), x\right) \]

      11. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(x \cdot x + \left(-1 - x \cdot x\right)\right)}\right), x\right) \]

      12. associate-+r- -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \color{blue}{\left(\left(x \cdot x + -1\right) - x \cdot x\right)}\right), x\right) \]

      13. fma-udef -0.0%

          \[\leadsto \mathsf{copysign}\left(-\left(\log \left(x - \mathsf{hypot}\left(1, x\right)\right) + \log \left(\color{blue}{\mathsf{fma}\left(x, x, -1\right)} - x \cdot x\right)\right), x\right) \]

      14. +-commutative -0.0%

          \[\leadsto \mathsf{copysign}\left(-\color{blue}{\left(\log \left(\mathsf{fma}\left(x, x, -1\right) - x \cdot x\right) + \log \left(x - \mathsf{hypot}\left(1, x\right)\right)\right)}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \mathsf{copysign}\left(-\color{blue}{\log \left(\mathsf{hypot}\left(1, x\right) - x\right)}, x\right) \]

   8. Taylor expanded in x around -inf 96.7%

      \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(-2 \cdot x\right)}, x\right) \]
   9. Step-by-step derivation

      1. *-commutative 96.7%

         \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   10. Simplified 96.7%

       \[\leadsto \mathsf{copysign}\left(-\log \color{blue}{\left(x \cdot -2\right)}, x\right) \]

   if -2 < x

   1. Initial program 35.0%

      \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
   2. Step-by-step derivation

      1. *-un-lft-identity 35.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

      2. *-commutative 35.0%

         \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

      3. log-prod 35.0%

         \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

      4. add-sqr-sqrt 24.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      5. fabs-sqr 24.0%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      6. add-sqr-sqrt 35.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

      7. +-commutative 35.2%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

      8. hypot-1-def 48.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

      9. metadata-eval 48.8%

         \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

   3. Applied egg-rr 48.8%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

   4. Taylor expanded in x around 0 70.2%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + 0, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\mathsf{copysign}\left(-\log \left(x \cdot -2\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{copysign}\left(x, x\right)\\ \end{array} \]

Alternative 12: 53.5% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary32 (copysign x x))

C

float code(float x) {
	return copysignf(x, x);
}

Julia

function code(x)
	return copysign(x, x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(x);
end

Derivation

1. Initial program 40.4%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Step-by-step derivation

   1. *-un-lft-identity 40.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(1 \cdot \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)\right)}, x\right) \]

   2. *-commutative 40.4%

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left(\left|x\right| + \sqrt{x \cdot x + 1}\right) \cdot 1\right)}, x\right) \]

   3. log-prod 40.4%

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right) + \log 1}, x\right) \]

   4. add-sqr-sqrt 18.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   5. fabs-sqr 18.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{\sqrt{x} \cdot \sqrt{x}} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   6. add-sqr-sqrt 30.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(\color{blue}{x} + \sqrt{x \cdot x + 1}\right) + \log 1, x\right) \]

   7. +-commutative 30.1%

      \[\leadsto \mathsf{copysign}\left(\log \left(x + \sqrt{\color{blue}{1 + x \cdot x}}\right) + \log 1, x\right) \]

   8. hypot-1-def 40.3%

      \[\leadsto \mathsf{copysign}\left(\log \left(x + \color{blue}{\mathsf{hypot}\left(1, x\right)}\right) + \log 1, x\right) \]

   9. metadata-eval 40.3%

      \[\leadsto \mathsf{copysign}\left(\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + \color{blue}{0}, x\right) \]

3. Applied egg-rr 40.3%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(x + \mathsf{hypot}\left(1, x\right)\right) + 0}, x\right) \]

4. Taylor expanded in x around 0 55.9%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x} + 0, x\right) \]

5. Final simplification 55.9%

   \[\leadsto \mathsf{copysign}\left(x, x\right) \]

Developer target: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t_0\right) + t_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary32
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

float code(float x) {
	float t_0 = 1.0f / fabsf(x);
	return copysignf(log1pf((fabsf(x) + (fabsf(x) / (hypotf(1.0f, t_0) + t_0)))), x);
}

Julia

function code(x)
	t_0 = Float32(Float32(1.0) / abs(x))
	return copysign(log1p(Float32(abs(x) + Float32(abs(x) / Float32(hypot(Float32(1.0), t_0) + t_0)))), x)
end

Reproduce

herbie shell --seed 2023279 
(FPCore (x)
  :name "Rust f32::asinh"
  :precision binary32

  :herbie-target
  (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))) x)

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))